Industrial Applications of Experimental Design

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Industrial Applications of Experimental Design

• John Borkowski
• Montana State University
• University of Economics and Finance
• HCMC, Vietnam

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Outline of the Presentation

1. Motivation and the Experimentation Process
2. Screening Experiments
3. 2k Factorial Experiments
4. Optimization Experiments
5. Mixture Experiments
6. Final Comments

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Motivation

• In industry (such as manufacturing,
  pharmaceuticals, agricultural, ), a common goal
  is to optimize production while maintaining
  quality and cost of production.
• To achieve these goals, successful companies
  routinely use designed experiments.
• Properly designed experiments will provide
  information regarding the relationship between
  controllable process variables (e.g., oven
  temperature, process time, mixing speed) and a
  response of interest (e.g. strength of a fiber,
  thickness of a liquid, color, cost).
• The information can then be used to improve the
  process making a better product more
  economically.

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Motivation

• The resulting economic benefits of using designed
  experiments include
• Improving process yield
• Reducing process variability so that products
  more closely conform to specifications
• Reducing development time for new products
• Reducing overall costs
• Increasing product reliability
• Improving product design

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The Experimentation Process
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Defining Experimental Objectives

• The first and most important step in an
  experimental strategy is to clearly state the
  objectives of the experiment.
• The objective is a precise answer to the question
  What do you want to know when the experiment is
  complete?
• When researchers do not ask this question they
  may discover after running an experiment that the
  data are insufficient to meet objectives.

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2. Screening Experiments

• The experimenter wants to determine which process
  variables are important from a list of
  potentially important variables.
• Screening experiments are economical because a
  large number of factors can be studied in a small
  number of experimental runs.
• The factors that are found to be important will
  be used in future experiments. That is, we have
  screened the important factors from the list.

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2. Screening Experiments

• Common screening experiments are
• Plackett-Burman designs
• Two-level full-factorial (2k) designs
• Two-level fractional-factorial (2k-p) designs
• Example Improve the hardness of a plastic by
  varying 6 important process variables. Goal
  Determine which of the six variables have the
  greatest influences on hardness.

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Example 1 Screening 6 Factors

• Response Plastic Hardness
• 
  Factor Levels
• Factors -1
  1
• (X1) Tension control Manual
  Automatic
• (X2) Machine 1
  2
• (X3) Throughput (liters/min) 10
  20
• (X4) Mixing method Single
  Double
• (X5) Temperature 200o
  250o
• (X6) Moisture level 20
  30

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Analysis of the Screening Design Data
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Interpretation of Results

• The most influential factor affecting plastic
  hardness is temperature, followed by throughput
  and machine type.
• To increase the hardness of the plastic, a higher
  temperature, higher throughput, and use of
  Machine type 2 are recommended.
• Tension control, mixing method, and moisture
  level appear to have little effect on hardness.
  Therefore, use the most economical levels of each
  factor in the process.
• A new experiment to further study the effects of
  temperature, throughput and machine type on
  plastic hardness is recommended for further
  improvement.

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2k Factorial Experiments

• A 2k factorial design is a design such that
• k factors each having two levels are studied.
• Data is collected on all 2k combinations of
  factor levels (coded as and - ).
• The 2k experimental combinations may also be
  replicated if enough resources exist.
• You gain information about interactions that was
  not possible with the Plackett-Burman design.

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Example 2 23 Design with 3 Replicates
(Montgomery 2005)

• An engineer is interested in the effects of
• cutting speed (A) (Low, High rpm)
• tool geometry (B) (Layout 1 , 2 )
• cutting angle (C) (Low, High degrees)
• on the life (in hours) of a machine tool
• Two levels of each factor were chosen
• Three replicates of a 23 design were run

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Experimental Design with Data

• Factors
• A cutting speed
• B tool geometry
• C cutting angle

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ANOVA Results from SASA cutting speed B
tool geometry C cutting angle
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Maximize Hours at B1 C1 A -1B tool
geometry C cutting angle A
cutting speed Layout 2 High
Low
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3. Optimization Experiments

• The experimenter wants to model (fit a response
  surface) involving a response y which depends on
  process input variables V1, V2, Vk.
• Because the exact functional relationship between
  y and V1, V2, Vk is unknown, a low order
  polynomial is used as an approximating function
  (model).
• Before fitting a model, V1, V2, Vk are coded as
  x1, x2, , xk. For example
• Vi 100
  150 200
• xi -1
  0 1

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4. Optimization Experiments

• The experimenter is interested in
• Determining values of the input variables V1, V2,
  Vk. that optimize the response y (known as the
  optimum operating conditions). OR
• Finding an operating region that satisfies
  product specifications for response y.
• A common approximating function is the quadratic
  or second-order model

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Example 3 Approximating Functions

• The experimental goal is to maximize process
  yield (y).
• By maximizing yield, the company can save a lot
  of money by reducing the amount of waste.
• A two-factor 32 experiment with 2 replicates was
  run with
• Temperature V1 Uncoded Levels 100o 150o
  200o
• x1 Coded
  Levels -1 0 1
• Process time V2 Uncoded Levels 6 8
  10 minutes
• x2 Coded
  Levels -1 0 1

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True Function y 5 e(.5x1 1.5x2)Fitted
function (from SAS)
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Predicted Maximum Yield (y) at x1 1 , x2
-1(or, Temperature 200o , Process Time 6
minutes)
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Central Composite Design Box-Behnken
Design (CCD)
(BBD)Factorial, axial, and
Centers of edges andcenter points
center points
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Example 4 Central Composite Design (Myers 1976)

• The experimenter wants to study the effects of
• sealing temperature (x1)
• cooling bar temperature (x2)
• polethylene additive (x3)
• on the seal strength in grams per inch of
  breadwrapper stock (y).
• The uncoded and coded variable levels are
• -? -1 0
  1 ?
  .
• x1 204.5o 225o 255o
  285o 305.5o
• x2 39.9o 46o 55o
  64o 70.1o
• x3 .09 .5 1.1
  1.7 2.11

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Example 4 Central Composite Design
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Ridge Analysis of Quadratic Model (using
SAS)Predicted Maximum at x1-1.01 x20.26
x30.68
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Further interpretation

• The predicted maximum occurs at coded levels of
  x1-1.01 x20.26 x30.68. These
  correspond to
• sealing temperature of 225o,
• cool bar temperature of 57.3o, and
• polyethelene additive of 1.51.
• Note how flat the maximum ridge is around this
  maximum. That implies there are other choices of
  sealing temperature, cool bar temperature, and
  additive that will also give excellent seal
  strength for the breadwrapper.
• Pick that combination that minimizes cost.

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5. Mixture Experiments

• Goal Find the proportions of ingredients
  (components) of a mixture that optimize a
  response of interest.
• 3-in-1 coffee mix has 3 components
  coffee,
• sugar, creamer. What are the proportions
  of
• the components that optimize the taste?
• Major applications formulation of food and drink
  products, agricultural products (such as
  fertilizers), pharmaceuticals.

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Mixture Experiments

• A mixture contains q components where xi is the
  proportion of the ith component (i1,2,, q)
• Two constraints exist 0 xi 1 and S
  xi 1

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Mixture Experiment Models

• Because the level of the final component can
  written as
• xq 1 (x1 x2
  xq-1)
• any response surface model used for
  independent factors can be reduced to a Scheffé
  model. Examples include

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Example of a 3-Component Mixture Design
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Analysis of a 3-component Mixture Experiment
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4-Component Mixture Experiment with Component
Level Constraints (McLean Anderson 1966)Goal
Find the mixture of Mg, NaNO3, SrNO3, and Binder
that maximize brightness of the flare.
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6. Final Comments

• Screening experiments
• 2k and 2k-p experiments
• Optimization experiments
• Mixture experiments
• Other applications
• Path of steepest ascent (descent) to locate a
  process maximum (minimum).
• Experiments with mixture and process variables.
• Repeatability and reproducability designs for
  statistical quality and process control studies.